Download - Short-Step Chebyshev Impedance Transformers
IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES, VOL. MTT-14, NO. 8, AUGUST, 1966
Short-Step Chebyshev Impedance Transformers
GEORGE L. MATTHAEI, FELLOW, IEEE
Absfract—Impedance transforming networks are described which
consist of short lengths of relatively high impedance transmission
line alternating with short lengths of relatively low impedance line.
The sections of transmission line are all exactly the same length
(except for corrections for fringing capacitances), and the lengths of
the line sections are typically short compared to a quarter wavelength
throughout the operating band of the transformer. Tables of designsare presented which give exactly Chebyshev transmission character-istics between resistive terminations having ratios ranging from1.5 to 10, and for fractional bandwidths ranging from 0.10 to 1.20.These impedance-transforming networks should have application
where very compact transmission-line or dielectric-layer im-pedance transformers are desired.
1. GENERAL
T
ABLES OF Chebyshev impedance-transformer
designs of lumped-element, low-pass-filter form
are presented in [1]. The structures used consist of
a ladder configuration of series inductances alternating
with shunt capacitances. The transmission character-
istic typically gives a sizable mismatch at dc, followed at
somewhat higher frequencies by a Chebyshev pass band
where the attenuation is very low, and at the upper edge
of the pass band, the transmission characteristic cuts off
in a manner typical of low-pass filters. It was also noted
in [1] that by techniques such as are used for designing
semi-lumped-element microwave filters [2], [3], it is
possible to design semi-lumped-element impedance-
transformers from the tables in [1]. Such procedures
are approximate but can give very good results if the
design is carefully worked out. Microwave semi-lumped-
eIement transformers of this type give performance
comparable to that of quarter-wave transformers, but
can be much smaller in size.
In this present discussion, the design of semi-lumped-
element impedance transformers consisting of short
sections of transmission line is treated on an exact basis.
Figure 1 shows a six-section transformer of the typeunder consideration. In this particular case the structure
is of coaxial form, and it consists of a cascade of line
sections having various impedances, each line section
having the same effective length 1. Such a transformer
Manuscript received December 7, 1965; revised May 4, 1966.This research was supported by the U. S. Army Electronics Labo-ratories, Ft. Monmouth, N. J., under Contract DA 36-039-AMC-00084(E).
The author is with the Department of Electrical Engineering,University of Californi?j Santa Barbara, Calif. He was formerly atStanford Research Institute, Menlo Park, Calif.
will give a good match between the line impedances ZO
and Z7 over a band of frequencies as indicated by the
transducer attenuation characteristic shown in Fig. 2.
The length 1 of the line sections in Fig. 1 is considerably
less than h~/4, where Am is the wavelength at the mid-
band frequency. This is also evident in Fig. 2, where it
will be seen that at the midband frequency of the lowest
pass band, the electrical length fl~ of the line sections is
considerably less than 7r/2. Unlike conventional quarter-
wave transformers, this type of impedance transformer
has maximum attenuation when the line sections are a
quarter-wavelength long i.e., when 0 = Tr/2). The attenu-
ation characteristic in Fig. 2 is periodic and has a maxi-
mum attenuation of L~l. This differs from the lumped-
element case discussed in [1], where the attenuation
characteristic is not periodic and the attenuation ap-
proaches infinity at high frequencies.
An interesting situation occurs when the length 1 of
the line sections is chosen to be A~/8. In that case, 19~ is
Tr/4, and for example, in Fig. 1, 21 = 22, Z3 = 24, 26’26.
The significance of this is that under this condition a
Chebyshev short-step transformer becomes identical
with a conventional Chebyshev quarter-wave trans-
former having half as many line sections. Thus, in order
to have the up and down variation of line impedances as
suggested in Fig. 1, 1 must be less than A~/8. The tables
presented herein are for the case where 1 = Am/l 6, and
results are given elsewhere for the case where 1=&/32
[15 ]. Tables for quarter-wave transformers (which as
explained above are equivalent to the case of 1= AJ8
herein) will be found in [4].
1 I. RESPONSE PARAMETERS
Let us now define various parameters for the response
characteristic in Fig. 2. Electrical length 0 will be used as
a frequency variable, where O is the electrical length of
each individual line section of the short-step trans-
former. The center frequency of the primary pass band
is defined by
& + eb 27rlem=—
2 ‘x’(1)
and the fractional bandwidth of the pass band is given by
ob – %~=—
Om(2)
1966 MATTI-LAEI: SHORT-STEP CHEBYSHEV IMPEDANCE TRANSFORMERS 373
~“’’’’”’”’’’’”’’’’’’”’’””?~q-~
F“= “ I-.. {‘7---=- ‘, w ‘3 ‘4- ‘5 ‘6 ‘7 1-/////////P/////.’//L”////////”//////////////// ‘-4
Fig. 1. A coaxial short-step impedance transformer designed tomatch between lines of impedance ZO and Z7. The sections of im-pedance Z1 to ZS are all of the same effective length 2.
ELECTRICAL LENGTH, o — rod,.”,
Fig. 2. The response characteristics of a typical short-step trans-former. The Darameter o is the electrical length of each of the linesections (see&Fig. 1).
where O. and f% are defined in Fig. 2. The attenuation
LM. at dc can be computed by
(?’+ 1)’ ~B~.idc = 1010glo ‘~ (3)
where r is the ratio of the terminating line impedance
Zn+l to the input line impedance 20.~Tormalizingthe line impedances so that Zo = 1 and
Zfi+l = r, the peak attenuation LAI can be computed by
the formula
LA, = ,(),Og,O ‘1 + ‘in)’: dB4Rin
(4)
where for n/2 even
r(zlz3 . . . zm/2_l)4Rin =
(Z2Z, . . . z.,2)’–(5)
and for n/2 odd
Rin = ‘2123 “ “ “ ‘“’2)E ~ (6)S’(Z2Z4 . . . 2.,,-,)’
Values for the normalized impedances ZI to Z~lZ arelisted in the tables accompanying this discussion.
Another important parameter of the response in Fig. 2
is the pass band attenuation ripple LA,. Values of LAr are
tabulated in Tables I through V for the various designs
which are tabulated in Tables VI through X. These data
are for the case where the line sections in Fig. 1 are
1=LJ16 in length. The Tables of LA, values are of
primary importance in determining which design should
be used for a given application. Assuming that 1=&/1 6
is desired, for the desired termination ratio r and frac-
tional bandwidth w, the corresponding value of LAr for
the n = 2 section case in Table I should be checked. If
this value of ripple is too large, then the table for n = 4
should be checked to see if n = 4 will result in a suffic-
iently small value of pass band ripple. If n = 4 still
gives too large a value of LA,, then n = 6 should be tried,
etc. In this manner the tables of LA, vs. r and w are used
to determine the number of line sections required in
order to give a sufficiently good degree of impedance
match across the desired operating frequency bancl.
The entire response of a given short-step transformerdesign can be computed by use of the mapping tech-niques discussed in Section V of this discussion. ‘l~his
may be of interest for cases where the regicms of high
attenuation in the response characteristic of a short-step
transformer are to be used for the purpose of filtering out
unwanted signals. The attenuation in the stop bands can
be very large. For example, for the case where 1=kJ16,
if n = 6 and r =5, the peak stop band attenuation L.41
runs about 45 dB for the entire range of w coverec[ by
the tables. If the line-section lengths are 1=&/32, n =6,
and r =5, then the peak attenuation LAI runs about
83 dB for all the values of w covered by the tables.
374 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES AUGUST
TABLE I
LA, vs. r ANII w FOR n=2 AND l=&~16
\
w0.1 0.2 0.3 0.4 0.6 0.8 1.0 1.2
r
1.52.02..5
R5.06.o7.08.0
1);
0.0016 0.0064 0.0141 0.0243 0.0502 0.07980.0049
0.10880.0191 0.0421 0.0724 0.1490
0.13420.2352 0.3187
0.0087 0.0344 0.07550.3906
0.1295 0.2645 0.4146 0.55780.0129 0.0509 0.1113
0.67970.1905
0.02180.3864 0.6009 ;; ;;3:
0.0855 0.18620.9725
0.3168 0.6334 0.97050.0310 0.1211 0.2625
1.53360.4439 0.8754 1.3230
0.04031.7239 2.0475
0.1570 0.3388 0.5695 1.1085 1.6545 2.13380.0497 0.1930 0.4145
2.51460.6928 1.3319 1.9655 2.5119 2.9403
0.0591 0.2289 0.4894 0.8134 1.5456 2.2573 2.86160.0686
3.33020.2647 0.5632 0.9312 1.7501 2.5319 3.1866
0.07803.6894
0.3003 0.6361 1.0462 1.9459 2.7908 3.4898 4.0220
TABLE II
LA, vs. r AND w FOR n=4 AND J=A~/16
\ !
\
w0.1 0.2 0.3 0.4 0.6 0.8 1.0 1.2
f’
TABLE III
LA, vs. r AND w FOR n=6 AND l=km/16
\
w
v
1.52.0
::24.05.06.0
:::
1;::
0.1 0.2 0.3 0.4 0.6 0.8 1.0 1.2
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
0.00000.00000.00000.00000.00010.00010.00020.00020.00020.00030.0003
0.00150.0017
0.00040.00120.00210.00310.00530.00750.00980.01210.01440.01680.0191
0.0022 0.00870.0067
0.02540.0259 0.0758
0.0121 0.0465 0.13540.0179 0.0688 0.19920.0302 0.1154 0.33100.0430 0.16320.0558
0.46350.2113 0.5943
0.0688 0.2594 0.72250.0818 0.3072 0.84780.0949 0.3547 0.97000.1079 0.4018 1.0892
TABLE IV
LiI, vs. r AND w FOR n=8 AND 1=?,J16
\
w
0.1 0.2 0.3 0.4 0.6 0.8 1.0r
1.2
1.52.02.5
:::5.06.07.0
n10.0
0.00000.00000.00000,00000.00000.00000.00000.00000 s 00000.00000.0000
0.00000.00000.00000.00000.00000.00000.00000.00000.00000.00000.0000
0.0000 0.00000.0000 0.00000.0000 0,00000.0000 0.00000.0000 0.00000.0000 0.00000.0000 0.00000.0000 0.00000.0000 0.00000.0000 0.00010.0000 0.0001
0.0000 0.0003 0.0020 0.00900.0001 0.0010 0.0061 0.02690.0002 0.0018 0.0110 0.04830.0003 0.0026 0.0163 0.07130.0004 0.0045 0.0274 0.11970.0006 0.0063 0.0389 0.16920.0008 0.0083 0.0506 0.21910.0010 0.0102 0.0624 0.26890.0012 0.0121 0.0742 0.31840.0014 0.0141 0.0860 0.36750.0016 0.0160 0.0978 0.4162
1966 MATT}iAEl: SHORT-STEP CHEBYSHEV IMPEDANCE TRANSFORMERS 375
TABLE V
LA, vs. r AND w FOR n=10 AND l=&J16
\
w
0.1 0.2 0.3 0.4 0.6 0.8 1.0 1.:17
M:::4.05.06.0
R9.0
10.0
0.0000 0.0000 0.0000 0.0000 0.0000 0.0000 0.00050.0000 0.0000
0.00310.0000 0.0000 0.0000
0.0000 0.00000.0001 0.0014 0. 00!)2
0.0000 0.0000 0.0000 0.0003 0.0025 0.01650.0000 0.0000 0.0000 0.0000 0.0000 0.0004 0.0038 0.02450.0000 0.0000 0.0000 0.0000 0.00000.0000
0.0007 0.00640.0000 0.0000
0.041[30.0000 0.0001 0.0009 0.0090 0.0586
0.0000 0.0000 0.0000 0.0000 0.0001 0.0012 0.0118 0.07610.0000 0.0000 0.0000 0.0000 .: 0.0001 0.0015 0.0145 0.09370.0000 0.0000 0.0000 0.0000 , ‘. ‘“ 0.0001 0.0018 0.0173 0.11140.0000 0.0000 0.0000 0.0000 k 0.0001 0.0021 0.0201 0.12’910.0000 0.0000 0.0000 0.0000 -y’& 0.0001 0.0023 0.0229 0.1467
—
TABLE VI
Z, vs. r AND w FOR n=2 AND 1=LJ16
Y
w0.1 0.2 0.3 0.4 0.6 0.8 1.0 1,2
r
1.83172.4782
1.82712.47092.99113.43664.19074.82975.394’15.90506.37526.8131
1.81962.45902.97603.41894.16874.80425.36555.87366.34126.7767
1.80942.44262.95523.39464.13854.76925.32625.83046.29456.7268
1.78142.39782.8982
1.74522.33932.82383.24033.94684.54635.07615.55595.99756.40896.7954
1.70332.27072,7361
1.65792.1956
3.00023.44734.20404.84525.41145.92406.39586.8351
2.63’963.02333.67614.23124.72225.16705.57665.95836.3170
3.32774.05554.67285.21805.71176.16616.5893
3.13713.81834.39684.90825.37155.79806.19536.5686
6.07.08.09.0
10.0 7.2479 7.2245 7.1860 7.1329 6.9870
TABLE VII-A
Z, VS. ?’ AND w FOR n=4 AND 1=LJ16
\
w
0.1 0.2r
1.7831 1.7864;:; 2.1446 2.15082.5 2.4008 2.40973.0 2.6037 2.61514.0 2.9205 2.9366.5.0 3.1679 3.1883
0.3 0.4 0.6 0.8 1.0 1.2
1.79182.16102.42442.63402,9633
1.79932.17522.444.82.66033.00063.27003.4961
1.81942.2142?.50162.73403.10623.4056
1.84402.26432.57592.83163.24813.5893
1.88642.36582.73563.04’773.57174.01424.40494.75895.08535.38!)85.6764
3.22233. 4.?866.0 3.3731 3.3976
7.0 3.5498 3.57838.0 3.7058 3.73799.0 3.8458 3.8816
10.0 3.9732 4.0126
3.66033.88464.08644.27114.4420
3.88414.14694.38604.60694.8131
3.6258 3.69283.7919 3.86793.9418 4.0266
4.45474.73795.00135.24854.0788 4.1722
TABLE VII-B
Zz VS. ?’ AND w FOR %=4 AND l= Am/16
\ ‘< 0“1 0“2 0.3 0.+ 0.6 0.8 1.0 1.2
;::2..53.04.05.0
0.52950.52140.53350.54940.58170.61150.63850.66310.68560.70640.7257
0.53120.52400.53690.55340.58710.61820.64630.67200.69560.71740.7378
0.53370.52770.5416
0.54100.53860.5557
0.55230.55470.5763
0.56820.57680.60410.63430.69360.74860.79960.84710.89170.93390.974’0
0.58950.60490.63880.67530.74630.81;!10.87320.93020.98381.0346
0.55920.59470.62760.65740.68470.70990.73320.7550
0.57620.61730.65540.69020.72220.75200.77980.8060
0.60110.65020.69570.73770.7766:; ;;;:
0.8797
6.0
i::9.0
10.0 1. 08:!9—
376 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES AUGUST
TABLE VIII-A
Z1 VS. i’ AND W I?OR n=6 AND l=bJ16
\
w0.1
r0.2 0.3 0.4 0.6 0.8 1.0 1.2
1.60061.822’71.97402.09072.26802.40272.51252.6057
1.62731.86622.03152.16022.3584.2.51122.63712.7450
1.52.0
1.58141.79161.93342.04192.20512.32792.4271
1.58511.79771.94142.05142.21732.34242.44362.5290
1.59151.8080
1.66631.93122.11802.26602.49782.68012.83292.9657
1.7184 1.78172.13172.39312.60902.96333.25643.51183.74103.95094.14594.3287
2.01992.23842.41482.69742.92.53
2..53.04.05.0
1.95482.06762.23812.36712.47182.5603
6.07.0
3.12003.29223.44783.59083.7238
2.51068.0 2.58319.0 2.6471
10.0 2.7047
2.6031 2.63742.70572.7673
2.6870 2.8401 3.08412.7593 2.9253 3.19152.8246 3.0029 3.2902
2.66882.7278
TABLE VIII-B
Zz VS. r AND W FOR n=6 AND l=&J16
\
w0.1 0.2 0.3 0.4 0.6 0.8 1.0 1.2
r
0.5026 0.5030 0.5036;:;
0.5046 0.5078 0.5133 0.5222 0.53610.4888 0.4896 0.4909 0.4928 0.4987 0.5086 0.5240
2.5 0.4897 0.4908 0.4926 0.4952 0.50350.5471
3.0 0.49400.5171
0.4953 0.49760.5382
0.5009 0.51120.5694
0.50490.5281 0.5543
0,5067 0.50970.5930
:::0.5141 0.5280
0.51570.5508
0.51790.5862 0.6384
0.5216 0.5269 0.5439 0.5720 0.61576.0 0.5257 0.5282
0.68020.5325
0.53480.5387 0.5585 0.5913 0.6427 0.7188
0.5376 0.5424 0.5495 0.5719 0.6091 0.6676 0.7546!:8 0.5432 0.5463 0.5516
0.55090.5594 0.5842 0.6255
0.55420.6909 0.7882
0.5600 0.5685 0.59551:::
0.6409 0.7128 0.82000.5580 0.5616 0.5678 0.5770 0.6062 0.6553 0.7335 0.8502
TABLE VIII-C
Zs VS. 7 AND W FOR n=6 AND l= Am/16
\
w
?’0.1 0.2 0.3 0.4 0.6 0.8 1.0 1.2
3.16243.83924.36504.81485.58075.23466.81507.34227.82858.28218.7088
3.15733.83214.35654.80515.56926.22176.80097.32707.81248.26518,6910
3.1488 3.1371 3.1045 3.06043.8203 3.80414.3424
3.7594 3.70014.3231 4.2700 4.2006
4.7892 4.7674 4.7078 4.63075.5504 5.5247 5.4550 5.36626.2006 6.1718 6.0942 5.99656.7778 6.7464 6.66227.3022
6.55747.2685 7.1786 7.0678
7.7860 7.7502 7.6553 7.53938.2373 8.1997 8.1002 7.97968.6618 8.6225 8.5188 8.3942
3.00543.62844.11824.54045.26455.88696.44176.94747.41517.85248.2646
2.93903.54504.02454.43955.15415.77036.32126.82427.29017.72648.1379
TABLE IX-.4
Zt vs. r AND w FOR n=8 AND l= Am/16
\
w
r
1.52.02.5
0.1 0.2 0.3 0.+ 0.6 0.8 1.0 1.2
1.46661.6218
1.42201.55’!91.64191.70701.80311.87401.93041.97742.01772.05322.0848
1.42561.56031.64861.71481.81261.8848
1.43181.56941.65991.72801.82871.90341.96302.01282.05572.09352.1272
1.44051.58251.67621.74691.85201.93011.99272.04522.09052.1304
1.50631.68241.80241.89502.03632.14442.2330
1.56271.77051.91582.03012.2081
1.63981.89452.07852.22692.46442.65592.81972.96463.09573.2163
1.72561.80461.92332.01252.0847
2.34752.46401.9424
1. 990.s2.03182.06802.1005
2.1456 2.30862.37512.43462.4887
2.56512.65512.73672.8116
2.19862.24562.287910.0 I 2.1662 3.3286
1966 MATTHAE[: SHORT-STEP CHEBYSHEV IMPEDANCE TRANSFORMERS 377
TABLE IX-B
Z, VS. Y AND W FOR n=8 AND 1=)wJ16
\ I
\
w0.1 0.2 0.3 0.4 0.6 0.8 1.0 1.2
r—-—
1.52.02.53.04.05.0
0.5215 0.5213 0.5210 0.5207 0.5200 0.5201 0.5223 0.528;10.5003 0.5003 0.5005 0.5007 0.5021 0.5056 0.5131 0.52800.4938 0.4941 0.4946 0.4954 0.4984 0.5046 0.5166 0.53860.4918 0.4922 0.4930 0.4942 0.4985 0.5071 0.5229 0.55130.4919 0.4926 0.4939 0.4958 0.5025 0.5149 0.5374 0.57690.4940 0.4949 0.4966 0.4991 0.5077 0.5235 0.55170.4966 0.4977 0.4997
0.60100.5027 0.5130
0.4992 0.50060.5317 0.5650 0.6233
0.5029 0.5064 0.5180 0.5394 0.5774 0.64400.5019 0.5034 0.5060 0.5098 0.52290.5044 0.5061 0.5089
0.54660.5132
0.5889 0.66340.5274 0.5534 0.5998 0.6817
0.5069 0.5086 0.5117 0.5163 0.5317 0.5598 0.6101 0.6991
6.07.08.0
1:::
TABLE IX-C
Zz VS. I’ AND W FOR n=8 AND l=&jJ16
\
w0.1 0.2 0.3 0.4 0.6 0.8 1.0 1.2
v—.—
1.52.02.53.04.05.06.07.08.09.0
10.0
2.94643.37913.68983.94164.34684.67394.95245.1972
2.94503.37813.68953.94.214.34914.67794.95815.2045
2.94263.37643.68903.94304.35284.68454.96765.2168
2.93913.37393.68823.94414.35794.69374.9807
2.92893.36653.68563.94704.37254.71995.0183
2.91433.35613.68233.95184.39434.75875.07375.3541
2.89543.34403.68073.96194.42874.81755.15645.4600
2.87253.33273.68553.98404.48654.91075.28455.62225.9326
5.23385.46165.66965.8619
5.28255.52105.73965.9421
5.41695.61725.8018
5.42595 62775.8139
5.44085.64525.8339
5.60845.84246.0598
5.73715.99326.2324
6.22126.4922
TABLE IX-D
Zd VS. t’ AND W FOR n=8 AND l=&/16
\ ,
\
w0.1
r0.2 0.3 0.4 0.6 0.8 1.0 1.2
——-—0.4146 0.4212 0.4304 0.4423 0.45710.4439 0.4527 0.4650 0.4810 0.50100.4743 0.4850 0.5000 0.5195 0.54400.5027 0.5151 0.5326 0.5553 0.58390.5533 0.5689 0.59080.5974
0.6194 0.65550.6158 0.6417 0.6756 0.7188
0.6366 0.6577 0.6873 0.72620.6721
0.77580.6957 0.7288 0.7724 0.8281.
0.7047 0.7306 0.7671 0.8151 0.87670.7349 0.7630 0.8027 0.85500.7631
0.92230.7934 0.8361 0.8925 0.965;!
0.41060.43860.46780.49520.54390.58630.62400.65800.68920.71810.7450
1.52.02.53.0
0.40960.43730.46620.49330.54160.58360.62080.65450.68530.71380.7404
0.41230.44080.47050.49830.54780.59090.62930.66390.69570.72510.7525
4.05.06.07.0
R10.0
TABLE X-A
Z, VS. Y AND W FOR n=10 AND l=im/16
\
w
7’
;.;
2.53.04.0
0.1 0.2 0.3 0.4 0.6 0.8 1.0 1.2
.—1.51731.70621. 839(II1.94392. 108,?2.237!92. 346!)2.44192. 526!12.60442.6757
1.3080 1.3112 1.3167 1.3246 1.3485 1.38541.3965 1.4012 1.4090 1.4202 1.4544
1.4394
1.4534 1.4589 1.46841.5078 1.5878
1.4820 1.5236 1.5893 1.68881.4954 1.5017 1.5125 1.5280 1.5756 1.6511 1.76681.5567 1.5642 1.5769 1.5954 1.6523 1.7437 1.88581.6013 1.6097 1.6241 1. 6+49 1.7092 1.8133 1.97711.6365 1.645’7 1.6614 1.6842 1.7547 1.86961.6657 1.6755 1.6924
2.05211.7168 1.7928 1.9171
1.6906 1.7010 1.71892.1163
1.7448 1.82561.7124 1.7’234 1.7421
1.9585 2.17291.7694 1.8546 1.9952 2.2236
1.7318 1.7432 1.7628 1.7913 1.8805 2.0283 2.2698
5.06.07.08.0
1%:
378 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES AUGUST
TABLE X-B
Zz W. r AND w FOR n=10 AND l=k/16
\
w
0.1 0.2 0.3r
0.4 0.6 0.8 1.0 1.2
1.52.02..53.04.05.0;::
8.09.0
10.0
0.55390.52770.5168
0.55270.52670.5161
0.5511 0.5467 0.5416 0.5372 0.53550.5255 0.5225 0.5199 0.51960.5152
0.52510.5134’ 0.5127 0.5157
0.5099 0.50900.5270
0.5100 0.5159 0.53220.51050.50510.50270.50170.50130.50130.5014
0.5050 0.5056 0.5093 0.5199 0.54460.5030 0.5049 0.5108 0.52530.5023 0,5052
0.55730.5130 0.5309
0.50210.5694
0.5060 0.51540.5024
0.5365 0.58080.5070 0.5180 0.5418 0.5916
0.5027 0.5082 0.5205 0.5469 0.60180.50020.5003 0.5008 0.5017 0.5032 0.5093 0.5230 0.5518 0.6115
TABLE X-C
Z, VS. Y AND W FOR n= 10 AND 1= L7J16
\w
0.1 (),2 0.3 0.4 0.6 0.8 1.0 1.2?’ ‘\
1.52.02.53.04.05.06.07.0
2.69112.9862
2.692.52.98873.18743.3414\3.57853.76203.91374.0440
2.69472.99283.1934
2.69782.99853,20163.35993.60493.79573.95404.0905
2.70573.01383.22453.39013.64893.85214.02194.1691
2.71523.03393.25573.43223.71133.93324.1204
2.73623.09143.35223.56713.91934.2094
3.18383.33673.57193.75363.90374.0325
3.34913.58953.77613.93054.0633
4.46054.2838 4.44594.4297 4.61454.5623 4.76874.6841 4.9113
4.68454.88825.07605.2510
8.09.0
10.0
4.14594.24774.3402
4.15894.26194.3557
4.18054.28594.3818
4.21114.31964.4185
4.29974.41784.5258
TABLE X-D
Z, VS. ‘r AND W FOR n= 10 AND l=ie)/16
\ 1
\
v,,
0.1 0.2 0.3 0.4 0.6 0.8 1.0Y
1.2
Jl—
1.5 0.39180.40480.41980.43390.45890.48020.49880.51530.53030.54400 5566
0.39270.40600.42120.43560.4610
0.39410.40800.42370.43850.46450.4868
0.40210.41900.43730.45430.4843
0.42210.44690.47190.49490.53540.57040.60130.62930.65490.67860.7007
2.02.53.04.05.0 0.4826
0.50160,51840.53370.54760.5605
0.49270.51290.53100,54730.56230.5762
0.51000.53250.55270.57100.58780.6034
6.07.08.09.0
10.0
0.5063I 0.52360.53930.55370.5669
0.69070.72260.75230.7801
TABLE X-E
Z, vS. I’ AND W ‘FOR rt=10 AND l= An,/16
\
w
r \
0.1 (),2 0.3 0.4 0.6 0,8 1.0 l,?
3.4290 3.35044.0160 3.91964.4982 4.38944.9210 4.80235.6541 5.51986,2882 6.14146.8550 6.69767.3720 7.20547.8503 7.67558.2975 8.11528.7187 8.5296
3,~61~3.81194.26914.67215.37435.98416.53067.03027.49327.92668.3354
1.52.02.53.04.05.06.07.0
:::10.0
3.58084.20504.71355.15765.92496.58687.1774
3.57314.19534.70245.14545.91096.57137.16067.69768,19408 65769.0943
3.56044.17944.68425.12535.88786.54577.13297.66808.16278.62489.0599
3.54284.15744.65915.09765.85606.51067.09497.62748.11978.57979.0129
3.49414.09664.58975.02135.76866.41416.99067.51628.00238.45658.8844
7.71558.21308,67769.11.51
1966 MATTHAEI: SHORT-STEP CHEBYSHEV IMPEDANCE TRANSFORMERS 379
III. THE IMPEDANCE TABLES
Line impedances for designs having line section
lengths 1=A~/16 are presented in Tables VI to X. (Ref-
erence [15 ] gives tables for .?= A/l 6 and 1= A/32 for ter-
mination ratios r = 1.5 to 20.) In all of these tables the
impedance values are normalized so that
ZIJ = 1, and Zn+l = r (7)
for the terminating line impedances on the left and right,
respective y. Since these circuits are anti metric, half of
the line impedances can be computed from the other
half. For this reason only half of the line impedances for
each design are included in the tables. The remaining
line impedances are easily computed by use of the
formula
rz{
—— —— . (8)j=(n/2)+1 to n Zn+l–j
It will be noted that there are tables for the cases of
n=2, 4, 6, 8, and 10 for the case of l= A~/16. For each
value of n there is a separate table for Zl, a separate
table for 22, etc., up to Zniz.
It is thus readily seen that a desired design is obtained
by using the tables of LA, in order to determine the
required value of n, and then selecting the impedance
values for half of the circuit from the corresponding
tables of impedances. The tables give the impedances for
half of the network, and the remaining impedances are
then obtained by use of (8).
IV. CORRECTIONS FOR FRINGING CAPACITANCES,
AND lLf ODIFIC.4T10N OF THE
IMPEDANCE VALLTtiS
The structure shown in Fig. 1 (and equivalent struc-
tures in waveguide) will have fringing capacitances
which occur at the steps between lines of different im-
pedance. The fringing fields at these !Steps may be repre-
sented by inserting lumped capacitances wherever a
junction between lines of different impedance occurs.
Reference [5] or [7] gives data from which the fringing
capacitance values for coaxial-line step discontinuities
can be obtained, and [6] or [7] gives corresponding data
for steps in waveguide. Having values for these fringing
capacitances, the designer can compensate for them by
making small adjustments in the physical lengths of the
various line sections in the short-step impedance trans-
former. Such corrections are important to make if best
performance is to be obtained.
One wayl to compensate for the fringing capacitances
is to replace the short sections of transmission lines in
Fig. 1 by lumped-element equivalent circuits. Figure
3(a) shows a length of transmission line, and the circuit
on the left in Fig. 3(b) shows an exact equivalent circuit
for this transmission line. On the right in Fig. 3(b) is
shown an approximate equivalent circuit for the trans-
mission line which is valid if the line length 1 is small
1 Another way is that given in [4], Sec. 6.08.
x~z .1.8 ~b==.~“=&
2CO=*
t < h/a
(c)
Fig. 3. Equivalent circuits for a length of transmission line. Theparameter Z is the characteristic impedance of the line, 1 is theline length, and v is the velocity of propagation.
compared to a quarter wavelength (say, less than A/8).
Figure 3(c) sho~vs another exact equivalent circuit for
the length of transmission line, and on the right is shown
yet another approximate equivalent circuit, which again
is accurate if 1 is small compared to a quarter wave-
length. For the structures under consideration, the
length of the line sections in the operating frequency
band of interest will be considerably less than a quarter
wavelength, and the approximate 7’- and ~r-equivalent
circuits on the right in Fig. 3 will be useful.
In Fig. 1 and in the other circuit designs tabu Iated
herein, the even-numbered line sections operate pre-
dominantly like shunt capacitors. It is therefore con-
venient to lump the fringing capacitances in with these
line sections. In order to compensate for the added ca-
pacitances due to the fringing fields, we may compute
the capacitance C. in the equivalent circuit on the right
in Fig. 3(c), for each even-numbered line section. Then
for each even-numbered line section we compute a com-
pensated length 1’ by making use of the equation
(9)
where CfL is the fringing capacitance at the left end of
the line section being treated, C$R is the fringing capacit-
ance at the right end, zk is the characteristic imped ante
of the line section, and v is the velocity of propagation.
The third term on the left in (9) will be recognized as
being the shunt capacitance of a length of transmission
line of impedance Z~ and of length 1’. Solving (9) for 1’
gives a compensated length 1’ which is slightly shorter
than the original design length 1.
To summarize, a recommended procedure for com-
pensating for the fringing effects in circuits such as Fig.
1 is to determine the fringing-capacitance values using
[5], [6], or [7], and then to solve for compensated ler~gths
1’ for the even-numbered line sections by solving (9).
380 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES AUGUST
(The capacitance of each line section, plus its fringing
capacitance, will then equal the total desired capaci-
tance for the given line section.) It appears reasonable
to ignore the small changes in inductance that result
from this procedure.
A special case occurs with respect to the fringing
capacitance and that exists at the junction between lines
20 and 21 in Fig. 1. This fringing capacitance may also
be compensated for by a somewhat different technique,
Besides the approximate equivalent circuits shown on
the right in Fig. 3, another reasonably accurate equiva-
lent circuit for a short length of transmission line is an
L-section circuit consisting of a series inductance L. and
a shunt capacitance C. given by the values
L=== (lo)v
(11)
where d is the length of the section of transmission line,
Z is the characteristic impedance of the line, and v is
again the velocity of propagation. Suppose the fringing
capacitance at the junction between 20 and Zl is Cf, If a
series inductance AL is placed beside this shunt-fringing
capacitance Cf, an L-section circuit can be obtained
having a characteristic impedance which will match the
terminating-line-impedance 2.. In terms of equations
this gives
‘0’ /%= & (12)
Solving for AL gives
AL = CfZo’. (13)
A practical way to obtain the desired inductance AL is
to increase the length of the transmission line 21 by an
amount d, which has the desired amount of inductance.
By (10), the inductance AL for a given value of d and line
impedance 21 is given by
ZldAL= —-.
v
Solving (13) and (14) for d gives
Vzf’cfd=—
21 “
(14)
(15)
Thus, in order to compensate for the fringing capaci-
tance Cf between lines 20 and 21 in the structure in Fig.
1 (or in any of the other designs tabulated herein), the
length of line section 21 should be increased from / to
l+d, where d is computed by use of (15).
V. THEORY BY WHICH THE TABLES
WERE OBTAINED
In some respects the synthesis procedure used for
obtaining the line-impedance values in the tables pre-
sented herein is similar to that discussed by Riblet [8]
for the design of Chebyshev, quarter-wave step trans-
formers. In discussing the synthesis procedure used
herein, we will make use of terminology similar to that
of Richards [9], Ozaki and Ishii [10], and Wenzel [11].
The transfer function for a step transformer such as
that in Fig. 1 ~vould involve transcendental functions. In
order to eliminate such functions, and to simplify the
synthesis problem, Richards [9] makes use of the map-
ping function
p = tanh ~ (16)
where s = u +jfl is the complex-frequency variable for
the transmission-line circuit, and P = a+ju is the fre-
quency variable of the mapped transfer function. The
parameter a used by Richards is defined by
(17)
where v is the velocity of propagation, and Q114 is the
radian frequency for which 1= A~/4. In order to apply
the mapping in (16), it is necessary that all of the trans-
mission lines in the structure be of length 1 or of some
integer multiple thereof. Using this mapping eliminates
the transcendental nature of the transfer and impedance
functions for transmission-line circuits, and makes pOS-
sible the use of well-known network synthesis techniques
which are commonly used for lumped-element circuits.
In terms of the characteristic impedance Z of a line
section, s, and a, the open-circuit impedance parameters
for a length of transmission line are given by
z211 = z~’2 = (18)
tanh ~
zz~~ = z~~ = (19)
sinh ~2
After applying the mapping in (16), these open-circuit
impedance parameters become
z211 = Z22 = —
P(20)
2<1 – p’ Z<(1 – p)(l + p)21’2 = 221 ==
P=~ (21)
P
1966 MATTliAEl: SHORT-STEP CHEBYSHEV IMPEDANCE TRANSFORMERS 381
These functions look like the open -circuit impedance
functions for a lumped-element circuit in terms of a com-
plex frequency variable ~, except fc,r the fact that the
transfer impedance functions Zlt =, 221 have one-half
order zeros of transmission at @= 1 amd p = — 1. Except
for this square root, the open-circuit impedances in (20)
and (21) would characterize a physically realizable
lumped-element circuit. Because of the nonphysical
realizability of the half-order zeros of transmission in
this circuit, Richards [9] has named the ~-plane equiva-
lent of a transmission line, a “unit element” (abbrevi-
ated U. E.). Thus the p-plane equivalent of the transmis-
sion line circuit in Fig. 1 is the cascade of unit elements
shown in Fig. 4.
Each of the unit elements in the circuit in Fig. 4
contributes to the transfer function, a half-order zero of
transmission at P = 1, and half-order zero of transmission
at P = — 1. Thus, for an n-section circuit of this sort,
there will be n half-order zeros of transrnission at P = 1
and also at @= — 1. In terms of an attenuation function,
the zeros of transmission become poles of attenuation,
and the transfer function for an n-section circuit of the
sort in Fig. 4 will be of the form
12h b.fn + b._@-’ + . . . b@’ + b,pz + b@ + bo—
En+l – (VI – p’)”
u(p)——(/1 – p’)” “ (22)
When this transfer function is factored so as to display
the locations of its poles and zeros, it takes the form
EO _ bn(P– PJ(t-PJ “ “ “ (P– P.). (23)
En~l – [j<(p - 1)(P + 1)1”
By use of conventional lumped-element-circuit network
synthesis techniques, along with sclme techniques in
Richards’ paper [9], it is possible to synthesize circuits of
the form Fig. 4 so as to have a prescribed transfer func-
tion of the form in (22) and (23).
The next problem to be treated is the problem of
synthesizing transfer functions of the form in (22) or
(23) so as to have a Chebyshev pass band. Let us first
investigate what the frequency response in the P plane
should look like. For frequencies s =jfl, the transforma-
tion in (16) becomes
(2’!)
Dropping the j’s and replacing aQ/2 Iby 0,
u = tan e. (25)
If we apply this mapping to the abscissa scale in Fig.
2, we will obtain a mapped transfer function of the form
in Fig. 5, where the function is sketched also for nega-
tive frequencies –a, for reasons of clarifying some of the
later discussion. Note that application of the malpping
(25) has eliminated the periodic character which the
transfer function had in Fig. 2.
The transfer function in Fig. 5 is in sclme respects
similar to the transfer function for analogous casets dis-
cussed in [1 ], and we shall make use of some similar
synthesis techniques. In [1], the desired transfer func-
tion was obtained by a mapping from the transfer
function of a conventional Chebyshev Iumperl-ele-
ment low-pass filter circuit. The transfer function for
a conventional three-reactive-element Chebyshev low-
pass filter circuit is shown in Fig. 6, where agai u the
transfer function has been sketched also for negative
frequencies. It will be noted that this transfer function
has three ripple-minima when negative frequencies are
included), while the transfer function in Fig. 5 has two
pass bands and six minima when negative frequencies are
included). Therefore, in order to obtain the transfer
function in Fig. 5 from that in Fig. 6, the transfer func-
Ro: 20 4U.E. u.E U E. LIE
H_-DuuIm
UE
Z2E, In, = Zz
Z3 z. Z5 26
zt
I.
Fig. 4. p-plane equivalent circuit of the transmission-linecircuit in Fig. 1.
Q1
7
/,
J“ TL Al LA(
LAdc
“ A A 1m -00 0 Wo 92
—-. uJ—
Fig. 5. The attenuation characteristic in Fig. 2 mappedto the p-plane by mapping Q = tan 0.
—-(of ~!_
Fig. 6. Transducer attenuation of a conventional t@ee-reactive-element Chebyshev low-pass filter. For mathematical purposesthe characteristic for negative frequencies is also shown.
382 IEEE TRANSACTIONS ON MICROWAVE THEORY AND TECHNIQUES AuGUST
tion in Fig. 6 must map into that in Fig. 5 twice. An-
other requirement here is that the multiple-order poles
of attenuation at infinity in the p‘ complex frequency
plane for the transfer function in Fig. 6 must map to
multiple-order poles at p = 1 and P = — 1 for the transfer
function in Fig. .5, as is required by the denominators of
(22) or (23). Some reflection on the matter will show
that a mapping function which accomplishes this is
(26)
where o’ is the sinusoidal frequency variable for the
conventional low-pass filter, u%’ is the cutoff frequency
of the conventional filter, and u is the frequency vari-
able for the circuit of the form in Fig. 4. The parameter
WO is as defined in Fig. 5, and it will be noted that the
point a’= O in Fig. 6 maps to the points co=UO and
w = —UO in Fig. 5. Extending the mapping in (26) to
complex frequencies by inserting P = jw and @‘ = jw’, we
obtain
(27)
The voltage attenuation function for a conventional
lumped-element, Chebyshev, low-pass filter circuit in
terms of complex frequencies p’ is of the form
El— = Cgp’q+ cQ_lpfel+ ..E,+ I
~ + c@’+ GO = u’(p’) (28)
and
= Cq(p’ — pl’)(p — p~’) . . . (p’ —pq’). (29)
The corresponding transducer power attenuation ratio
function is of the form
P’=’ = g’ U’(Z’)U’(– p’)
PL’(30)
where P’a.ail is the available power of the generator, PL’
is the power delivered to the load, U’ (p’) is as defined in
(28), and g’ is a real, constant multiplier. For sinusoidal
frequencies, the transducer attenuation in dB is given
by
P’ avail
LA = 10 Ioglo —P’n+l r=~w
(31)
and it is this function which is sketched in Fig. 6. The
corresponding transducer-power attenuation ratio func-
tion for the circuit in Fig. 4 is of the form
P.T.il _ gu(p)u(–p)(~ – pz)n “
(32)PL –
It is readily seen that by inserting the mapping function
in (27) into (30), a transducer attenuation ratio function
of the desired form in (32) will be obtained. It should be
noted that the power of the numerator polynomial in
(32) will then be double the power of the numerator
polynomial in (30). As a result, for the corresponding
voltage attenuation ratio functions in (28) and (22),
n=2q.
Closed-form expressions giving the locations of the
zeros of transducer power attenuation ratio functions of
the form in (30) for conventional Chebyshev low-pass
filters are available [12 ]. However, it will be more con-
venient to ~vork with the reflection coefficient function
N’(p’)F’(p’)= —
u, ~p~ “ (33)
Putting this function in power reflection coefficient
form, we obtain
Prefl N’ (p’) 11” ( – p’)—— = r’(p’) r’(–p’) = —— (34)P.V.il U’(p’) u’(-p’)
where P,efl is the reflected power. Closed-form expres-
sions for the location of both the poles and zeros of po~ver
reflection coefficient functions as in (34) for conven-
tional Chebyshev, lumped-element, low-pass filter cir-
cuits are also available [13 ].
By inserting the mapping in (27) into (34), the corre-
sponding power reflection coefficient function
* = r(f)r(–p) =A’(p) AT( – p)
(35)aval u(p)~j(–t)
for the circuit in Fig. 4 is obtained. After throwing away
the right-half-plane poles and zeros of this function, and
eliminating any canceling factors that may have been
introduced by the mapping process, the voltage reflec-
tion coefficient function
N(p)r(p) = —
[T(j)
K(p – jcd.) (p + ju.) (p – jq) (p + jqJ . . ..
(P– PI)(P– PJ(P-P3)(P-P4) “ “ “
(36)
is obtained. It is of interest to note that the denominator
of (36) has the same roots as does the numerator of (23),
while the numerator of (36) has all of its roots located on
the jti-axis. The voltage reflection coefficient function,
(36), is of most immediate use in the synthesis procedure
so that the formation of the function in (35) may be
bypassed. Thus, only the poles and zeros of (34) which
map to the left-half plane or to the jo-axis are included
in the mapping process. The mapping is accomplished
most directly by solving (27) for p to obtain
p=+d
j% ’cm’A + p’(37)
–jwJA + p’ “
Thus, for each pole or zero of (31), (37) will yield two
poles and zeros. From this array of mapped poles and
zeros, we select all of the poles that are in the left-hand
plane, and use them to form (36). Mapping the zeros of
(31) will yield double zeros on the jco axis of the @-plane,
and we select one zero from each of these double zeros to
form the numerator of (36).
1966 MATTHAEI: SHORT-STEP CHEBYSHEV IMPEDANCE TRANSFORMERS 383
It is still necessary to specify how the parameter A is
to be determined. Now 19. and 8b can be determined in
terms of f?~ and the fractional bandwidth w by use of
()&=o. +
()8.=0,,, 1–;.
(38)
(39)
By inserting (25) into (26), and using the fact that
~’ =UZ’ \when o =Ob, we may solve for .4 and obtain
~ + tan28b.4=— —
tan26’b— tan2 ~0(40)
m-here~
6J0 == tan O.. (41)
Since co’ = –am when O=da, by use of (25), (26),
(40), we can solve for tan 00 to obtain
tan 80
d
(tan @b)’ [l+ (tan I%)’]+ (tan 6U)2[1 + (tan Oh)’]——
2+ (tan e.) 2+ (tan- @b) 2
Using (42) in (40), we may then compute A.
and
(42)
Other transfer-function parameters may be obtained
by mapping from the transfer function for a conven-
tional lumped-element low-pass Chebyshev filter. This
transfer function is
–=1+’c0sh2(’cOs’’-’3’43)P.~.il
Pouh
where
[
LA,
1e=antilog ~ — 1 . (44)
At the maximum attenuation point in Fig. 2, 0 = 7r/2,~ = tan ~/2 = a, and by (26), oJ’/w~’ =.4. Therefore,
where the fact that % = 2g has been made use of. The
maximum attenuation L~l is this ratio expressed in
decibels. Equation (45) will also be useful in determining
the constant multiplier K in (36). Since I I’ I is equal to
K ~vhen $ equals j ~,
and by use of (45) and (46), we obtain
(46)
2 Note that 00 is not the same as b’~ (see Fig. 2.)
Using these results the reflection coefficient function in
(36) can be completely specified.
After the reflection coefficient function is obtained
with its numerator and denominator polynomials in
factored form, these polynomials must next be multi-
plied out. Then assuming ZO = 1, the input impedance
function Zi. in Fig. 4 is given by
(48)
Next the circuit is synthesized by removing one unit
element at a time by successive application of (9) cjf [9].
In this manner the impedance values 21, Zz, etc., asso-
ciated with the unit elements in Fig. 4, are obtained, and
these impedance values are also the characteristic im-
pedances of the line sections in Fig. 1. Since the zeros of
the reflection coefficient function in (36) are all on the
ja-axis, the net~vork must be either symmetric or anti-
metric (in this case it is antimetric) [14], and the sec-
ond half of the network can be computed from the first
half. For this reason it is necessary only to compute half
of the impedance values, and the other half can be com-
puted by (8).
ACKNOWLEDGMENT
Acknowledgment is due V. H. Sagherian, whc] pre-
pared the computer program for computing the tables of
designs in this paper.
[1]
[2]
[3]
[4][5]
[6]
[7]
[8]
[9]
[10]
[11]
REFERENCES
G. L. Matthaei, “Tables of Chebyshev impedance-transformingnetworks of low-pass filter form, ” Proc. IEEE, vol. 52, pp. 939–963, August 1964.Harvard Radio Research Labs. staff, Very High l%equenc~r Tech-niques, vol. 2. New York: McGraw-Hill, 1947, pp. 653–6!;4 and685–692.G. L. Matthaei, L. Young, and E. M. T. Jones, lJesigtr of Micro-wave Filte~s, Iwpedance-Matching Netwo~ks, and Couplinf Struc-tures. New York: McGraw-Hill, 1964, ch. 7.—, ibid. 3, ch. 6.T. R. Whinnerv. H. W. Tamieson. and T. E. Robbins. “Coaxial-~ine discontinu~~ies, ” P;oc. IRE, ‘vol. 32, pp. 695-709, Novem-ber 1944.N. Marcuvitz, W’a~eguide Handbook, vol. 10, M. I .T. Rad. Lab.Ser. New York: McGraw-Hill, 1951, p. 309.G. L. Matthaei, L. Young, and E. M. T. Jones, op. cit. [3], Figs.5.07-2 and 5.07-11.H. J. Rib let, “General synthesis of quarter-wave impedancetransformers, ” IRE Trans. on Microwave Theory and Techniques,vol. MTT-5, pp. 36–43, January 1957.P, I. Richards, “Resistor-transmission-lines circuits, ” Proc. IRE,vol. 36, pp. 217–2201. February 1948.H. Ozaki and J. Ishu, “Synthesis of transmission-line networksand the design of UHF filters. ” IRE Tram. on Circuit Theory,vol. CT-2, p;. 325–336, Decem’ber 1955.
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“Theoretical limitations on the broadband matching ofarbit’mry impedances, ” ~. Franklin Inst., vol. 249, pp. 57-84and 139–154, January and February 1950.E. A. Guillemin, Synthesis of Passive Networhs. New York:Wiley, 1957, ch. 11.G. L. hIatthaei, “Tables of short-step impedance transformerdesigns. ” These tables cover impedance ratios from r= 1.5 to 20,for bandwidths from w= 0.1 to 1.2 for line lengths 1= x~/116 andL= kJ32. Copies ar~ available from: AD I Auxiliary PublicationsProject, Photoduphcation Service, Library of Congress, ‘Wash-ington, 25, D. C. To order cite Document 8901. Charge is $3.75for photoprints; or $2.00 for 3.5 mm microfilm. Make checkspayable to: Chief, Photoduplication Service, Librawy of Congress,